A note on $k$-wise oddtown problems

TL;DR

This paper investigates generalized oddtown problems involving multiple set families with intersection parity conditions, establishing tight bounds on the size of these families for various parameters.

Contribution

It provides new upper bounds on the size of set families satisfying specific intersection parity conditions, extending classical oddtown results and proving these bounds are optimal.

Findings

Established bounds for $m$ in terms of $n$ for different $t$ and $k$

Proved the bounds are tight and best possible

Generalized classical oddtown problem results

Abstract

For integers $2 \leq t \leq k$, we consider a collection of $k$ set families $\mathcal{A}_j: 1 \leq j \leq k$ where $\mathcal{A}_j = \{ A_{j,i} \subseteq [n] : 1 \leq i \leq m \}$ and $|A_{1, i_1} \cap \cdots \cap A_{k,i_k}|$ is even if and only if at least $t$ of the $i_j$ are distinct. In this paper, we prove that $m =O(n^{ 1/ \lfloor k/2 \rfloor})$ when $t=k$ and $m = O( n^{1/(t-1)})$ when $2t-2 \leq k$ and prove that both of these bounds are best possible. Specializing to the case where $\mathcal{A} = \mathcal{A}_1 = \cdots = \mathcal{A}_k$, we recover a variation of the classical oddtown problem.